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I agree with Mike that you shouldn't NEED this, BUT the answer is yes. The monad $T$ arises as coming from the adjunction $U:C^T \to C:F$, where $C^T$ is its category of Eilenberg-Moore algebras. Sinc …

I believe that your theorem is false. Suppose that for every small category $C$ I make an arbitrary choice of representing object for each isomorphism class. Then, the fullsubcategory on these chosen …

In general, the counit of an adjunction is an isomorphism if and only if the right-adjoint is fully faithful (dually the unit is an iso iff the left-adjoint is fully-faithful). So, your question is ea …

Lax-pseudo limits, bilimits, or weak-2-limits- choose your terminologoy- don't use non-invertible $2$-cells in their definitions or universal properties. (If you ask for genuine lax or oplax limits, i …

The general definition of sheaf, is much more general than a local homeomorphism over a space- however, it is sheaves of this later kind that came about first. In the general framework, you have a sma …

Todd already gave a beautiful answer. However, I will try to give a quick answer myself in simpler language: Since $\eta:F \Rightarrow \Delta_H$ is a limiting cone, for each object $A$, the induced m …

This question is related to this one: Which limits are preserved by a reflective left-adjoint? (And in fact may be seen as a special case, but I think it merits its own question). Suppose that $C$ i …

See Leinster's "Higher Operads, Higher Categories" Example 2.1.10. Maps of multicategories which come from monoidal categories are precisely the same as lax monoidal functors.

Let $F:C \to D$ be a functor. Then, for all $c \in C_0$ we have an induced functor $F/c:C/c \to D/Fc$. Suppose that for each $c$, $F/c$ has a right adjoint. There's surely a name for such a functor. W …

I am interested in the following situation: Suppose that $i:C \to D$ is a functor, $C$ does not necessarily have a terminal object, and for each object $c$ of $C,$ the induced functor $$C/c \to D/i( …

If there is an adjunction between a category $C$ and $D$ such that the left adjoint $$L:C \to D$$ is faithful (but not full), can one describe the image of $L$ in terms of the co-unit of the adjunctio …

This question is closely related to this one: Which limits are preserved by prolongation of presheaves? Suppose that $r:C \hookrightarrow D$ is a full and faithful functor and $l:D \to C$ is a left-a …

Is the category of topological spaces locally presentable? n-lab claims that it is not locally FINITELY presentable, but how about for some larger cardinal? Here I really mean the 1-category of topolo …

The category of Grothendieck topoi and (equivalence classes of) geometric morphisms. For example, if $A$ is the classifying topos for abelian groups, the geometric morphisms from $Set$ to $A$ are in b …

I'll attempt to answer some of your questions. First off, 1) and 3) are equivalent. This is because the bicategory of fractions in 3) is the Morita bicategory of topological groupoids, which is equiva …